The hadith of the man who prayed badly is often referenced, at least among Sunni schools, due to its clear descriptions of prayer; it describes, step-by-step, each of the mandatory forms of the regular salat.

The common narrations of this hadith do not, however, describe what mistake (or mistakes) were made in the original prayer to actually earn the reprimand "Go back and pray, for you have not prayed."

Is there any information available regarding these mistakes? Authentic sources are, of course, preferable, but anything from the hadith literature which sheds light on this event would be appreciated.

Allah's Messenger (sallallahu 'alaihi wa sallam) entered the mosque and a person followed him. The man prayed and went to the Prophet and greeted him. The Prophet (sallallahu 'alaihi wa sallam) returned the greeting and said to him, Go back and pray, for you have not prayed. The man went back prayed in the same way as before, returned and greeted the Prophet who said, Go back and pray, for you have not prayed. This happened thrice. The man said, By Him Who sent you with the Truth, I cannot offer the prayer in a better way than this. Please, teach me how to pray. The Prophet (sallallahu 'alaihi wa sallam) said, When you stand for Prayer say Takbir and then recite from the Holy Qur'an (of what you know by heart) and then bow till you feel at ease. Then raise your head and stand up straight, then prostrate till you feel at ease during your prostration, then sit with calmness till you feel at ease (do not hurry) and do the same in all your prayers.

The word feel at ease meaning الطمأنينة in Arabic is repeated 3 times in the hadith.
I think the prayer was in a hurry when praying.
More lecture in arabic about this hadith and the 'educational side' on it here

It was narrated that Rifa'ah bin Rafi' said: "While the Messenger of
Allah () was sitting with us around him, a man came in, turned towards
the Qiblah and prayed. When he had finished his prayer, he came and
greeted the Messenger of Allah () and the people with Salam. The
Messenger of Allah () said to him: 'And also to you. Go and pray, for
you have not prayed.' So he went and prayed, and the Messenger of
Allah () started watching him, and he (the man) did not know what was
wrong with it. When he had finished the prayer, he came and greeted
the Messenger of Allah () and the people with salam. The Messenger of
Allah () said to him: 'And also to you. Go and pray, for you have not
prayed.' He repeated it two or three times, then the man said: 'O
Messenger of Allah, what is wrong with my prayer?' The Messenger of
Allah () said: 'The prayer of any of you is not complete unless he
performs wudu properly as enjoined by Allah, the Mighty and Sublime.
So he should wash his face, his arms up to the elbows, and wipe his
head, and (wash) his feet up to the ankles. Then he should magnify
Allah (SWT) and praise Him and glorify Him.'" - (One of the narrators)
Hammam said: "I heard him say: 'He should praise Allah and glorify Him
and magnify Him." He said: "I heard both of them." -"He (the Prophet
()) said: 'He should recite whatever is easy for him of the Quran that
Allah has taught him and permitted him in it (the prayer). Then he
should say the Takbir and bow until his joints settle and he is
relaxed. Then he should say: 'Sami Allahu liman hamidah (Allah hears
the one who praises Him)' and stand up straight until his backbone is
straight (and at ease). Then he should say Takbir and prostrate until
he has placed his face firmly on the ground." "I heard him say: his
forehead, until his joints settle and he is relaxed. Then he should
say the Takbir and sit up until his backbone is straight (and at
ease). Then he should prostrate until he has placed his face firmly on
the ground and he is relaxed. If he does not do that then he has not
completed his prayer." (Sahih)